A268295 Terms at square positions in Pascal's triangle when in flattened form.

1, 1, 2, 1, 5, 15, 1, 126, 10, 220, 715, 15, 12870, 17, 27132, 15504, 1540, 1144066, 1, 5311735, 80730, 475020, 54627300, 32, 1037158320, 46376, 254186856, 854992152, 82251, 131282408400, 820, 151532656696, 2481256778, 260932815, 6973199770790, 1, 67327446062800, 636763050, 841392966470, 96926348578605, 367290, 15033633249770520, 4582116, 2044802197953900

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Formula

a(A001109(n)) = 1 and a(3*A001109(n)) = 1 for n>=0, where A001109(n)^2 is a triangular number.

Example

Pascal's triangle begins:
(1),
(1), 1,
1, (2), 1,
1, 3, 3, (1),
1, 4, 6, 4, 1,
1, (5), 10, 10, 5, 1,
1, 6, 15, 20, (15), 6, 1,
1, 7, 21, 35, 35, 21, 7, 1,
(1), 8, 28, 56, 70, 56, 28, 8, 1,
1, 9, 36, 84, (126), 126, 84, 36, 9, 1,
1, 10, 45, 120, 210, 252, 210, 120, 45, (10), 1,
1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1,
1, 12, 66, (220), 495, 792, 924, 792, 495, 220, 66, 12, 1,
1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, (715), 286, 78, 13, 1,
1, 14, 91, 364, 1001, 2002, 3003, 3432, 3003, 2002, 1001, 364, 91, 14, 1,
1, (15), 105, 455, 1365, 3003, 5005, 6435, 6435, 5005, 3003, 1365, 455, 105, 15, 1,
1, 16, 120, 560, 1820, 4368, 8008, 11440, (12870), 11440, 8008, 4368, 1820, 560, 120, 16, 1, ...
in which the terms of this sequence (shown enclosed in parenthesis) are found at positions n^2, for n>=0, in the flattened form of the triangle.
Positions of where 1's occur in this sequence: [0, 1, 3, 6, 18, 35, 105, 204, 612, 1189, 3567, 6930, 20790, 40391, 121173, 235416, ..., A001109(n), 3*A001109(n), ...], where A001109(n)^2 is a triangular number.

Prog

(PARI) {a(n) = my(t=floor(sqrt(2*n^2+1)-1/2)); binomial(t, n^2 - t*(t+1)/2)}
for(n=0, 50, print1(a(n), ", "))
(PARI) /* Positions of Where 1's occur in this sequence: */
for(n=0, 10000, t=floor(sqrt(2*n^2+1)-1/2); if(binomial(t, n^2 - t*(t+1)/2)==1, print1(n, ", ")))

Crossrefs

Cf. A001109, A007318.

Sequence in context: A174317 A144237 A350742 * A063635 A192264 A144054

Adjacent sequences: A268292 A268293 A268294 * A268296 A268297 A268298

Keyword

nonn

Author

Paul D. Hanna, Feb 21 2016

Status

approved